$a$ – the coefficient of $X^2$.

$b$ – the coefficient of $X$.

$c$ – the constant term.

$a$ – the coefficient of $X^2$.

$b$ – the coefficient of $X$.

$c$ – the constant term.

- Let's examine the parameter $a$ and ask: Is the function upward or downward facing?
- Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
- Let's find the points of intersection with the $X$ axis by substituting ($Y=0$).
- Let's draw a coordinate system and first mark the vertex of the parabola.

Then, let's examine if the function is smiling or crying and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

\( y=-2x^2+3x+10 \)

Let's present the quadratic function and understand the meaning of each parameter in it.

$y=ax^2+bx+c$

$a$ – the coefficient of $X^2$

- -determines if the parabola will be a maximum or minimum parabola (sad or smiling) - determines the steepness – its width.

$a$ must be different from 0.

If $a$ is positive – the parabola is a minimum parabola – smiling

If $a$ is negative – the parabola is a maximum parabola – sad

The larger $a$ is – the narrower the function will be and vice versa.

$b$ - the coefficient of $X$

can be any number

indicates the position of the parabola along with $C$.

determines the slope of the function at the intersection point with the $Y$ axis

(not relevant to the material)

$c$ – the constant term

is not dependent on $X$

can be any number

indicates the position of the parabola along with $X$

determines the y-intercept

responsible for the vertical shift of the function

Wonderful. Now, let's move on to plotting the quadratic function.

- Let's examine the parameter $a$ and ask: is the function upward or downward facing?
- Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
- Let's find the points of intersection with the $X$ axis by substituting ($Y=0$).
- Let's draw a coordinate system and first mark the vertex of the parabola.

Then, let's examine if the function is smiling or sad and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

**Let's see an example -**

In the following function:

$Y=-4X^2+4x+3$

- $a=-4$, negative. Therefore, the function is downward facing.
- Let's find the vertex of the parabola:

$X=\frac{-4}{-4*2}=\frac{-4}{-8}=\frac{1}{2}$

$y=-4*\frac{1^2}{2}+4*\frac{1}{2}+3$

$y=4$

Vertex of the parabola $(\frac{1}{2}, 4)$

- Let's find the intersection points with the $X$ axis

Substitute

$Y=0$

and we get:

$-4X^2+4x+3=0$

$X=-\frac{1}{2},1\frac{1}{2}$

- Draw a coordinate system, mark relevant points, and draw logically:

Test your knowledge

Question 1

\( y=x^2 \)

Question 2

\( y=x^2+10x \)

Question 3

\( y=x^2-6x+4 \)

Related Subjects

- Quadratice Equations and Systems of Quadraric Equations
- Quadratic Equations System - Algebraic and Graphical Solution
- Solution of a system of equations - one of them is linear and the other quadratic
- Intersection between two parabolas
- Word Problems
- Properties of the roots of quadratic equations - Vieta's formulas
- Ways to represent a quadratic function
- Various Forms of the Quadratic Function
- Standard Form of the Quadratic Function
- Vertex form of the quadratic equation
- Factored form of the quadratic function
- The quadratic function
- Quadratic Inequality
- Parabola
- Symmetry in a parabola
- Finding the Zeros of a Parabola
- Methods for solving a quadratic function
- Completing the square in a quadratic equation
- Squared Trinomial
- Families of Parabolas
- The functions y=x²
- Family of Parabolas y=x²+c: Vertical Shift
- Family of Parabolas y=(x-p)²
- Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)